Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

中文翻译：

---

连续域分配流

分配流表示用于图上的上下文数据标记（分类）的一类动态模型。我们推导出了一种新的分配流参数化，揭示了底层信息几何如何分别引发分配正则化和逐步执行明确决策的两个过程，这两个过程在求解流时无缝交互。我们的结果能够将分配流的主要部分描述为相对于底层信息几何的黎曼梯度流。我们考虑相应势的连续域公式，并在求解受简单凸约束的线性椭圆偏微分方程 (PDE) 序列方面开发一种新算法。